Should I be able to be nominated at the award sites I run?
Of course! Writing and Webmistressing are totally different.
Well, since you can't nominate or vote for yourself at the sites your run, it's not like you can skew the votes.
No! You wouldn't want people to play favorites because it's your site.
Other. My thoughts are in comments.
Depending on your answers, I might let myself be nominated at the SunnyD Awards next round, too.